On non-abelian extensions of Leibniz algebras

0.21 MB·

by  Jiefeng Liu

On non-abelian extensions of Leibniz algebras∗
Jiefeng Liu1, Yunhe Sheng2 and Qi Wang2
1Department of Mathematics, Xinyang Normal University,
Xinyang 464000, Henan, China
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1 2Department of Mathematics, Jilin University, Changchun 130012, Jilin, China
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2 Email: jfl[email protected], [email protected], [email protected]

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u
J

6 Abstract
2
In this paper, first we classify non-abelian extensions of Leibniz algebras by the second

] non-abelian cohomology. Then, we construct Leibniz 2-algebras using derivations of Leibniz
T
algebras, and show that under a condition on the center, a non-abelian extension of Leibniz
C
algebras can be described by a Leibniz 2-algebra morphism. At last, we give a description of
. non-abelianextensionsintermsofMaurer-CartanelementsinadifferentialgradedLiealgebra.
h
t
a
m 1 Introduction
[

Eilenberg and Maclane developed a theory of non-abelian extensions of abstract groups in the
2
1940s [12], leading to the low dimensional non-abelian group cohomology. Then there are a lot of
v
9 analogous results for Lie algebras [1, 18, 19]. Abelian extensions of Lie algebras can be classified
8 bythe secondcohomology. Non-abelianextensionsofLiealgebrascanbe describedbysomelinear
1 maps regarded as derivations of Lie algebras. This result is generalized to the case of super Lie
3
algebrasin [2, 15], andto the case ofLie algebroidsin[7]. In[13], the authorgavea descriptionof
0
non-abelianextensionsofLie algebrasinterms ofthe Deligne groupoidofadifferentialgradedLie
.
1 algebra(DGLA for short). In [25], a non-abelian extensionof the Lie algebrag by the Lie algebra
0
ad
6 h is explained by a Lie 2-algebra morphism from g to the strict Lie 2-algebra h−→Der(h), where
1 Der(h) is the Lie algebra of derivations on h. See [5] for more information about a Lie 2-algebra,
: which is a categorification of a Lie algebra.
v
i The notion of a Leibniz algebra was introduced by Loday [22, 23], which is a noncommutative
X
generalization of a Lie algebra. Central extensions of Leibniz algebras are studied from different
r aspects [10, 11, 17]. Abelian extensions of Leibniz algebrasis studied in [9]. Non-abelian cohomol-
a
ogyandextensions bycrossedmodules ofLeibniz algebrasarestudiedin[8,16,20]. Associatedto
a Leibniz algebra,there is adifferentialgradedLie algebrastructure onthe gradedvectorspaceof
the cochain complex, which plays an important role in studying cohomology and deformations of
Leibniz algebras,see [6, 14] for details.
0Keywords: Leibniz algebras, non-abelian extensions, non-abelian cohomology, Leibniz 2-algebras, Maurer-
Cartan elements
0MSC:17B99,55U15
∗ThisresearchissupportedbyNSFC(11471139) andNSFofJilinProvince(20140520054JH).
1
In this paper, we study non-abelian extensions of Leibniz algebras. First we introduce the
second non-abelian cohomology H2(g,h) of a Leibniz algebra g with the coefficients in a Leibniz
algebra h, by which we classify non-abelian extensions of g by h. Then we construct two Leibniz
2-algebras using left derivations and right derivations of Leibniz algebras. One is a natural gener-
alization of the Lie 2-algebra associated to derivations of a Lie algebra; The other can be used to
describe non-abelian extensions of Leibniz algebras that satisfy a condition on centers. Whether
there is a Leibniz 2-algebra associated to derivations of Leibniz algebras, that could be used to
describeallthenon-abelianextensionsofLeibnizalgebras,likethecaseofLiealgebras,remainsto
be an interesting problem. Finally, we generalize the result in [13] to the case of Leibniz algebras.
We constructaDGLAassociatedtoLeibniz algebrasg andh, anddescribenon-abelianextensions
of g by h by its Maurer-Cartanelements.
The paper is organizedas follows. In Section 2, we study properties of left derivations DerL(g)
and right derivations DerR(g) of a Leibniz algebra g, and construct a semidirect product Leib-
niz algebra DerL(g)⊕DerR(g). Then we introduce the second non-abelian cohomology of Leib-
niz algebras, by which we classify non-abelian extensions of Leibniz algebras (Theorem 2.8). In
Section 3, first we recall some basic definitions regarding Leibniz 2-algebras and morphisms be-
tween them. Then we construct Leibniz algebras Π(h) and Ξ(h) associated to a Leibniz algebra
h, which are subalgebras of DerL(h)⊕DerR(h). We go on constructing strict Leibniz 2-algebras
(h,Π(h),(adL,adR),l ) and (h,Ξ(h),(adL,adR),l ). The former is a natural generalization of the
2 2
Lie2-algebraassociatedtoderivationsofaLiealgebra,andthelattercanbeusedtodescribesome
non-abelian extensions of g by h. More precisely, we show that there is a one-to-one correspon-
dence between non-abelian extensions ˆg of g by h with Z(h)= Z(ˆg)∩h and morphisms from g to
(h,Ξ(h),(adL,adR),l ) (Theorem 3.12). In Section 4, we construct a DGLA (L,[·,·] ,∂), which is
2 C
asub-DGLAof(C(g⊕h),[·,·] ,∂),whereg⊕histhe Leibnizalgebradirectsumofgandh. Then
C
we show that the equivalence relation for non-abelian cohomology coincides with the equivalence
relation for Maurer-Cartanelements in the DGLA (L,[·,·] ,∂) (Theorem 4.7).
C
2 Classification of non-abelian extensions of Leibniz alge-
bras in terms of non-abelian cohomology
A Leibniz algebra is a vector space g, endowed with a linear map [·,·] :g⊗g−→g satisfying
g
[x,[y,z] ] =[[x,y] ,z] +[y,[x,z] ] , ∀ x,y,z ∈g. (1)
g g g g g g
This is in fact a left Leibniz algebra. In this paper, we only consider left Leibniz algebras.
Denote by Z(g) the left center of g, i.e.
Z(g)={x∈g|[x,y] =0, ∀y ∈g}. (2)
g
A representation of the Leibniz algebra (g,[·,·] ) is a triple (V,l,r), where V is a vector
g
space equipped with two linear maps l : g −→ gl(V) and r : g −→ gl(V) such that the following
equalities hold:
l =[l ,l ], r =[l ,r ], r ◦l =−r ◦r , ∀x,y ∈g. (3)
[x,y]g x y [x,y]g x y y x y x
The Leibniz cohomology of g with coefficients in V is the cohomology of the cochain complex
Ck(g,V) = Hom(⊗k+1g,V),(k ≥ 0) with the coboundary operator ∂ : Ck−1(g,V) −→ Ck(g,V)
2
defined by
k
∂c(x ,...,x ) = (−1)i+1l (c(x ,...,x ,...,x ))+(−1)k+1r (c(x ,...,x ))
1 k+1 xi 1 i k+1 xk+1 1 k
Xi=1
b
+ (−1)ic(x ,...,x ,...,x ,[x ,x ] ,x ,...,x ). (4)
1 i j−1 i j g j+1 k+1
1≤i<Xj≤k+1
b
We denote by DerL(g) and DerR(g) the set of left derivations and the set of right derivations
of g respectively:
DerL(g) = {D∈gl(g)|D[x,y] =[Dx,y] +[x,Dy] ,∀x,y ∈g},
g g g
DerR(g) = {D∈gl(g)|D[x,y] =[x,Dy] −[y,Dx] ,∀x,y ∈g}.
g g g
It is easy to see that for all x ∈ g, adL : g −→ g, which is given by adL(y) = [x,y] , is a left
x x g
derivation; adR :g−→g, which is given by adR(y)=[y,x] , is a right derivation.
x x g
Lemma 2.1. If x∈Z(g), then for all DL ∈DerL(g), we have DLx∈Z(g).
Proof. It follows from
[DLx,y]=DL[x,y]−[x,DLy]=0, ∀ x∈Z(g).
For all D ,D ∈ gl(g), [D ,D ] denotes their commutator, i.e. [D ,D ] = D D −D D . By
1 2 1 2 1 2 1 2 2 1
straightforwardcomputations, we have
Lemma 2.2. For all x∈g, DL,DL,DL ∈DerL(g) and DR ∈DerR(g), we have
1 2
(i) [DL,DL]∈DerL(g);
1 2
(ii) [DL,DR]∈DerR(g);
(iii) [DL,adL]=adL , [DL,adR]=adR , [adL,DR]=adR .
x DLx x DLx x DRx
Corollary 2.3. With the above notations, DerL(g) is a Lie algebra, and DerR(g) is a DerL(g)-
module, where the representation ρ is given by
ρ(DL)(DR)=[DL,DR], ∀DL ∈DerL(g),DR ∈DerR(g).
Denote by Der(g)=DerL(g)⊕DerR(g).
ViewDerL(g)asaLeibnizalgebra,then(ρ,0)isarepresentationofDerL(g)onDerR(g). Thus,
we have the semidirect product Leibniz algebra.
Lemma 2.4. (Der(g),[·,·] ) is a Leibniz algebra, where the bracket operation [·,·] on Der(g) is
s s
given by
[(DL,DR),(DL,DR)] =([DL,DL],[DL,DR]), ∀ DL ∈DerL(g), DR ∈DerR(g). (5)
1 1 2 2 s 1 2 1 2 i i
Definition 2.5. (1) Let g, h, ˆg be Leibniz algebras. A non-abelian extension of Leibniz algebras
is a short exact sequence of Leibniz algebras:
0−→h−→i ˆg−p→g−→0.
We say that ˆg is a non-abelian extension of g by h.
3
(2) A splitting of ˆg is a linear map σ :g→g such that ◦σ =id.
p
(3) Twoextensionsofg byh, ˆg andˆg , arebsaid tobeisomorphic ifthereexistsaLeibniz algebra
1 2
morphism θ :ˆg −→ˆg such that we have the following commutative diagram:
2 1
0 −→ h −i→2 gˆ2 −p→2 g −→ 0
θ (6)
(cid:13)  (cid:13)
0 −→ (cid:13)(cid:13)h −i→1 gˆy1 −p→1 (cid:13)(cid:13)g −→ 0.
Let ˆg be a non-abelian extension of g by h, and σ : g → g a splitting. Define ω : g⊗g → h,
l:g−→gl(h) and r :g−→gl(h) respectively by
b
ω(x,y) = [σ(x),σ(y)] −σ[x,y] , ∀x,y ∈g, (7)
gˆ g
l (β) = [σ(x),β] , ∀x∈g, β ∈h, (8)
x gˆ
r (α) = [α,σ(y)] , ∀y ∈g, α∈h. (9)
y gˆ
Givena splitting, we haveˆg∼=g⊕h, and the Leibniz algebrastructure on ˆg can be transferred
to g⊕h:
[x+α,y+β] =[x,y] +ω(x,y)+l β+r α+[α,β] . (10)
(l,r,ω) g x y h
Proposition 2.6. (g⊕h,[·,·] ) is a Leibniz algebra if and only if l,r,ω satisfy the following
(l,r,ω)
equalities:
l [α,β] = [l α,β] +[α,l β] , (11)
x h x h x h
r [α,β] = [α,r β] −[β,r α] , (12)
x h x h x h
[l α+r α,β] = 0, (13)
x x h
[l ,l ]−l = adL , (14)
x y [x,y]g ω(x,y)
[l ,r ]−r = adR , (15)
x y [x,y]g ω(x,y)
r (r (α)+l (α)) = 0, (16)
y x x
l ω(y,z)−l ω(x,z)−r ω(x,y) = ω([x,y] ,z)−ω(x,[y,z] )+ω(y,[x,z] ). (17)
x y z g g g
Proof. Assume that (g⊕h,[·,·] ) is a Leibniz algebra. By
(l,r,ω)
[x,[α,β] ] =[[x,α] ,β] +[α,[x,β] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we deduce that (11) holds. By
[α,[β,x] ] =[[α,β] ,x] +[β,[α,x] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we deduce that (12) holds. By
[x,[y,α] ] =[[x,y] ,α] +[y,[x,α] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we deduce that (14) holds. By
[x,[α,y] ] =[[x,α] ,y] +[α,[x,y] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
4
we deduce that (15) holds. By
[x,[y,z] ] =[[x,y] ,z] +[y,[x,z] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we deduce that (17) holds. By
[α,[x,β] ] =[[α,x] ,β] +[x,[α,β] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we obtain that l [α,β] = [α,l β] −[r α,β] . Combining with (11), we deduce that (13) holds.
x h x h x h
By
[α,[x,y] ] =[[α,x] ,y] +[x,[α,y] ] ,
(l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω) (l,r,ω)
we obtain that l r −r +r r =adR . Combining with (15), we deduce that (16) holds.
x y [x,y]g y x ω(x,y)
Conversely, if (11)-(17) hold, it is straightforward to see that (g⊕h,[·,·] ) is a Leibniz
(l,r,ω)
algebra. The proof is finished.
Note that (11) means that l ∈DerL(h), and (12) means that r ∈DerR(h).
x x
Definition 2.7. A non-abelian 2-cocycle on g with values in h is a triple (l,r,ω) of linear maps
l : g → DerL(h), r : g → DerR(h) and ω : g⊗g → h satisfying Eqs. (13)-(17). We denote by
Z2(g,h) the set of non-abelian 2-cocycles.
Two 2-cocycles (l1,r1,ω1) and (l2,r2,ω2) are equivalent, if there exists a linear map ϕ:g→h
such that for all x,y ∈g, the following equalities hold:
l1−l2 = adL , (18)
x x ϕ(x)
r1 −r2 = adR , (19)
x x ϕ(x)
ω1(x,y)−ω2(x,y) = l2ϕ(y)+r2ϕ(x)+[ϕ(x),ϕ(y)] −ϕ([x,y] ). (20)
x y h g
The secondnon-abelian cohomology H2(g,h) is thequotient of Z2(g,h)by theabove equivalence
relation.
Theorem 2.8. Let (g,[·,·] ) and (h,[·,·] ) be Leibniz algebras. Then non-abelian extensions of g
g h
by h are classified by the second non-abelian cohomology H2(g,h).
Proof. Let ˆg be a non-abelianextensionof g by h. By choosing a splitting σ :g−→ˆg, we obtain
1
a 2-cocycle (l1,r1,ω1). First we show that the cohomological class of (l1,r1,ω1) does not depend
on the choice of splittings. In fact, let σ and σ be two different splittings. Define ϕ:g−→h by
1 2
ϕ(x) :=σ (x)−σ (x). By lj(β) = [σ (x),β] and rj(α) = [α,σ (y)] , j = 1,2, it is obvious that
1 2 x j gˆ y j gˆ
(18) and (19) hold. Furthermore, we have
ω1(x,y) = [σ (x),σ (y)] −σ [x,y]
1 1 gˆ 1 g
= [σ (x)+ϕ(x),σ (y)+ϕ(y)] −σ [x,y] −ϕ([x,y] )
2 2 gˆ 2 g g
= [σ (x),σ (y)] +[σ (x),ϕ(y)] +[ϕ(x),σ (y)] +[ϕ(x),ϕ(y)] −σ [x,y] −ϕ[x,y]
2 2 gˆ 2 gˆ 2 gˆ h 2 g g
= σ [x,y] +ω2(x,y)+l2ϕ(y)+r2ϕ(x)+[ϕ(x),ϕ(y)] −σ [x,y] −ϕ([x,y] )
2 g x y h 2 g g
= ω2(x,y)+l2ϕ(y)+r2ϕ(x)+[ϕ(x),ϕ(y)] −ϕ([x,y] ),
x y h g
which implies that (20) holds. Thus, (l1,r1,ω1) and (l2,r2,ω2) are in the same cohomological
class.
5
Now we go on to prove that isomorphic extensions give rise to the same element in H2(g,h).
Assume that gˆ and gˆ are isomorphic extensions of g by h, and θ :gˆ −→gˆ is a Leibniz algebra
1 2 2 1
morphism such that we have the commutative diagram (6). Assume that σ : g −→ gˆ and
2 2
σ :g−→gˆ are two splittings. Define σ′ :g−→gˆ by σ′ =θ−1◦σ . Since ◦θ = , we have
1 1 2 2 2 1 p1 p2
◦σ′(x)=( ◦θ)◦θ−1◦σ (x)=x.
p2 2 p1 1
Thus, σ′ is a splitting of gˆ. Define ϕ :g−→h by ϕ (x)=σ′(x)−σ (x). Since θ−1 :g →g is
2 2 θ θ 2 2 1 2
a morphism of Leibniz algebras,we have
θ−1[σ (x),α] =[θ−1σ (x),α] =[ϕ (x)+σ (x),α] =l2α+adL α.
1 gˆ1 1 gˆ2 θ 2 gˆ2 x ϕθ(x)
On the other hand, since θ| =id, we have
h
θ−1[σ (x),α] =[σ (x),α] =l1α.
1 gˆ1 1 gˆ1 x
Therefore, we have
l1−l2 =adL . (21)
x x ϕθ(x)
Similarly, we have
r1 −r2 =adR . (22)
x x ϕθ(x)
For all x,y ∈g, on one hand, we have
θ−1[σ (x),σ (y)] = [θ−1σ (x),θ−1σ (y)]
1 1 gˆ1 1 1 gˆ2
= [ϕ (x)+σ (x),ϕ (y)+σ (y)]
θ 2 θ 2 gˆ2
= [ϕ (x),ϕ (y)] +l2ϕ (y)+r2ϕ (x)+[σ (x),σ (y)]
θ θ h x θ y θ 2 2 gˆ2
= [ϕ (x),ϕ (y)] +l2ϕ (y)+r2ϕ (x)+σ [x,y] +ω2(x,y).
θ θ h x θ y θ 2 g
On the other hand, we have
θ−1[σ (x),σ (y)] = θ−1(σ [x,y] +ω1(x,y))=ϕ [x,y] +σ [x,y] +ω1(x,y).
1 1 gˆ1 1 g θ g 2 g
In conclusion, we have
ω1(x,y)−ω2(x,y)=l2ϕ (y)+r2ϕ (x)+[ϕ (x),ϕ (y)] −ϕ [x,y] . (23)
x θ y θ θ θ h θ g
By (21)-(23), we deduce that (l1,r1,ω1) and (l2,r2,ω2) are in the same cohomologicalclass.
Conversely,giventwo2-cocycles(l1,r1,ω1)and(l2,r2,ω2)representingthesamecohomological
class, i.e. there exists ϕ : g −→ h such that (18)-(20) holds, we show that the corresponding
extensions (g⊕h,[·,·] ) and (g⊕h,[·,·] ) are isomorphic. Define θ :g⊕h−→g⊕h
(l1,r1,ω1) (l2,r2,ω2)
by
θ(x+α)=x−ϕ(x)+α.
Then, we can deduce that θ is a Leibniz algebra morphism such that the following diagram com-
mutes:
0 −→ h −i→2 g⊕h(l2,r2,ω2) −p→2 g −→ 0
θ
(cid:13)  (cid:13)
0 −→ (cid:13)(cid:13)h −i→1 g⊕h(ly1,r1,ω1) −p→1 (cid:13)(cid:13)g −→ 0.
We omit details. This finishes the proof.
6
3 Another approach to non-abelian extensions of Leibniz
algebras
The notion of strongly homotopy (sh) Leibniz algebras, or Lod -algebras was introduced in [21],
∞
see also [4, 26] for more details. In [24], the authors introduced the notion of a Leibniz 2-algebra,
whichisthecategorificationofaLeibnizalgebra,andprovethatthecategoryofLeibniz2-algebras
and the category of 2-term Lod -algebras are equivalent.
∞
Definition 3.1. A Leibniz 2-algebra V consists of the following data:
d
• a complex of vector spaces V :V −→V ,
1 0
• bilinear maps l :V ×V −→V , where 0≤i+j ≤1,
2 i j i+j
• a trilinear map l :V ×V ×V −→V ,
3 0 0 0 1
such that for any w,x,y,z ∈V and m,n∈V , the following equalities are satisfied:
0 1
(a) dl (x,m)=l (x,dm),
2 2
(b) dl (m,x)=l (dm,x),
2 2
(c) l (dm,n)=l (m,dn),
2 2
(d) dl (x,y,z)=l (x,l (y,z))−l (l (x,y),z)−l (y,l (x,z)),
3 2 2 2 2 2 2
(e ) l (x,y,dm)=l (x,l (y,m))−l (l (x,y),m)−l (y,l (x,m)),
1 3 2 2 2 2 2 2
(e ) l (x,dm,y)=l (x,l (m,y))−l (l (x,m),y)−l (m,l (x,y)),
2 3 2 2 2 2 2 2
(e ) l (dm,x,y)=l (m,l (x,y))−l (l (m,x),y)−l (x,l (m,y)),
3 3 2 2 2 2 2 2
(f) the Jacobiator identity:
l (w,l (x,y,z))−l (x,l (w,y,z))+l (y,l (w,x,z))+l (l (w,x,y),z)
2 3 2 3 2 3 2 3
−l (l (w,x),y,z)−l (x,l (w,y),z)−l (x,y,l (w,z))
3 2 3 2 3 2
+l (w,l (x,y),z)+l (w,y,l (x,z))−l (w,x,l (y,z))=0.
3 2 3 2 3 2
We usually denote a Leibniz 2-algebraby (V ,V ,d,l ,l ), or simply by V. A Leibniz 2-algebra
1 0 2 3
is a strict Leibniz 2-algebraif l =0.If l is skewsymmetric,we obtainLie 2-algebras, see [5]
3 2
for more details.
Lemma 3.2. For a Leibniz 2-algebra (V ,V ,d,l ,l ), we have
1 0 2 3
l (l (x,m),y)+l (l (m,x),y)=0, ∀ x,y ∈V ,m∈V . (24)
2 2 2 2 0 1
Definition 3.3. Let V and V′ be Leibniz 2-algebras, a morphism f from V to V′ consists of
• linear maps f :V −→V′ and f :V −→V′ commuting with the differential, i.e.
0 0 0 1 1 1
f ◦d=d′◦f ;
0 1
7
• a bilinear map f :V ×V −→V′,
2 0 0 1
such that for all x,y,z ∈L , m∈L , we have
0 1
l′(f (x),f (y))−f l (x,y) = d′f (x,y),
2 0 0 0 2 2
 l′(f (x),f (m))−f l (x,m) = f (x,dm), (25)
2 0 1 1 2 2
 l′(f (m),f (x))−f l (m,x) = f (dm,x),
2 1 0 1 2 2

and
f (l (x,y,z))+l′(f (x),f (y,z))−l′(f (y),f (x,z))−l′(f (x,y),f (z))
1 3 2 0 2 2 0 2 2 2 0
−f (l (x,y),z)+f (x,l (y,z))−f (y,l (x,z))−l′(f (x),f (y),f (z))=0. (26)
2 2 2 2 2 2 3 0 0 0
Let (h,[·,·] ) be a Leibniz algebra. Define ∆(h)⊂Z(h) by
h
∆(h)=span{[α,α] |α∈h}. (27)
g
Define Π(h)⊂Der(h) by
Π(h)={(DL,DR)∈Der(h)|DLα+DRα∈∆(h), ∀α∈h}. (28)
Lemma 3.4. For any β ∈∆(h) and DR ∈DerR(h), we have DRβ =0.
Proof. For all α ∈ h and DR ∈ DerR(h), we have DR[α,α] = 0, which implies that for all
h
β ∈∆(h), DRβ =0.
Proposition 3.5. Π(h) is a sub-Leibniz algebra of (Der(h),[·,·] ).
s
Proof. By Lemma 3.4, for any β ∈ ∆(h) and (DL,DR) ∈ Π(h), DLβ ∈ ∆(h). Thus, for all
(DL,DR),(DL,DR)∈Π(h), we have
1 1 2 2
[(DL,DR),(DL,DR)] α = [DL,DL]α+[DL,DR]α
1 1 2 2 s 1 2 1 2
= DL(DLα+DRα)−(DLDLα+DRDLα)∈∆(h),
1 2 2 2 1 2 1
which implies that [(DL,DR),(DL,DR)] ∈Π(h). Thus, Π(h) is a sub-Leibniz algebra.
1 1 2 2 s
It is obvious that for all α∈h, (adL,adR)∈Π(h).
α α
(adL,adR)
Consider the complex h −→ Π(h), and define l by
2
l ((DL,DR),(DL,DR)) = [(DL,DR),(DL,DR)] , for (DL,DR)∈Π(h),
2 1 1 2 2 1 1 2 2 s i i
 l2((DL,DR),α) = DLα, for (DL,DR)∈Π(h), α∈h, (29)
 l (α,(DL,DR)) = DRα, for (DL,DR)∈Π(h), α∈h.
2

Proposition 3.6. With the above notations, (h,Π(h),(adL,adR),l ) is a strict Leibniz 2-algebra,
2
where l is given by (29).
2
Proof. By Lemma 2.2, we have
(adL,adR)l ((DL,DR),α) = (adL,adR)(DLα)=adL +adR
2 DLα DLα
= [DL,adL]+[DL,adR]
α α
= [(DL,DR),(adL,adR)α] =l ((DL,DR),(adL,adR)α),
s 2
8
which implies that Condition (a) in Definition 3.1 holds.
Similarly, we have
dl (α,D)−l (dα,D) = adL +adR −([adL,DL]+[adL,DR])
2 2 DRα DRα α α
= adL =0,
DLα+DRα
which implies that Condition (b) in Definition 3.1 holds.
Condition (c) follows from
l ((adL,adR),β)=l (α,(adL,adR))=[α,β] .
2 α α 2 β β h
By Proposition 3.5, Condition (d) holds naturally.
For all (DL,DR),(DL,DR)∈Π(h), we have
1 1 2 2
l (α,l ((DL,DR),(DL,DR)))−l (l (α,(DL,DR)),(DL,DR))−l ((DL,DR),l (α,(DL,DR)))
2 2 1 1 2 2 2 2 1 1 2 2 2 1 1 2 2 2
=[DL,DR]α−DRDRα−DLDRα=−DR(DLα+DRα)=0,
1 2 2 1 1 2 2 1 1
which implies that Condition (e ) in Definition 3.1 holds.
1
Similarly, Conditions (e ) and (e ) also hold. Since l = 0, Condition (f) holds naturally. The
2 3 3
proof is finished.
It is well-known that when h is a Lie algebra, there is a strict Lie 2-algebra (h,Der(h),ad,l ),
2
where Der(h) is the set of derivations of h, and l is given by
2
l (D ,D ) = [D ,D ], for D ,D ∈Der(h),
2 1 2 1 2 1 2 (30)
(cid:26) l2(D,α) = −l2(α,D)=Dα, for D ∈Der(h), α∈h.
When h is a Lie algebra, it is obvious that ∆(h)=0. Therefore, Π(h) reduces to
Π(h)={(D,−D)|D ∈Der(h)}.
Furthermore,thestrictLeibniz2-algebragiveninProposition3.6reducestoaLie2-algebra. Define
f :Der(h)−→Π(h) by
0
f :D 7−→(D,−D), ∀D ∈Der(h).
0
About the relation between the two Lie 2-algebras,we have
Proposition 3.7. Let h be a Lie algebra, then (f ,f = id,f = 0) is an isomorphism from the
0 1 2
Lie 2-algebra (h,Der(h),ad,l ) to (h,Π(h),(adL,adR),l ).
2 2
Proof. It is obvious that f and f commute with the differential. Furthermore, we have
0 1
f l (D ,D ) = ([D ,D ],−[D ,D ])=[(D ,−D ),(D ,−D )] =l (f (D ),f (D )),
0 2 1 2 1 2 1 2 1 1 2 2 s 2 0 1 0 2
f l (D,α) = f (Dα)=Dα=l (f (D),f (α)).
1 2 1 2 0 1
Thus, (f ,f =id,f =0) is an isomorphism between Lie 2-algebras.
0 1 2
Remark 3.8. From the above result, we see that the Leibniz 2-algebra associated to a Leibniz
algebra given in Proposition 3.6 is a natural generalization of the Lie 2-algebra associated to a Lie
algebra. However, we find that we can construct another Leibniz 2-algebra associated to a Leibniz
algebra, which plays an important role when we study non-abelian extensions of Leibniz algebras.
9
Define a subspace Ξ(h)⊂Der(h) by
Ξ(h)={(DL,DR)∈Der(h)|DLα+DRα∈Z(h), for all α∈h, and DRZ(h)=0}. (31)
It is obvious that for all α∈h, (adL,adR)∈Ξ(h).
α α
Proposition 3.9. Ξ(h) is a sub-Leibniz algebra of (Der(h),[·,·] ).
s
Proof. For all (DL,DR), (DL,DR)∈Ξ(h) and α,β ∈h, by Lemma 2.1, we have
1 1 2 2
[[(DL,DR),(DL,DR)] α,β] = [[DL,DL]α+[DL,DR]α,β]
1 1 2 2 s h 1 2 1 2 h
= [DL(DLα+DRα),β] −[DLDLα+DRDLα,β]
1 2 2 h 2 1 2 1 h
= 0.
Furthermore, by Lemma 2.1 and the fact that DRZ(h)=0, for all α∈Z(h), we have
2
[DL,DR]α=DLDRα−DRDLα=0.
1 2 1 2 2 1
Thus, [(DL,DR),(DL,DR)] α∈Ξ(h), which implies that Ξ(h) is a sub-Leibniz algebra.
1 1 2 2 s
Corollary 3.10. Ξ(h) is a right ideal of (Der(h),[·,·] ).
s
Similar as Proposition 3.6, we have
Proposition 3.11. With the above notations, (h,Ξ(h),(adL,adR),l ) is a strict Leibniz 2-algebra,
2
where l is given by (29) for all (DL,DR)∈Ξ(h), (DL,DR)∈Ξ(h).
2 i i
Define Pr :Der(h)−→DerL(h) and Pr :Der(h)−→DerR(h) by
L R
Pr (DL,DR)=DL, Pr (DL,DR)=DR.
L R
Theorem 3.12. Letˆg=(g⊕h,[·,·] )beanon-abelian extensionofgbyhwithZ(h)=Z(ˆg)∩h.
(l,r,ω)
Then, (l,r,ω) give rise to a Leibniz 2-algebra morphism (f ,f ,f ) from g to the Leibniz 2-algebra
0 1 2
(h,Ξ(h),(adL,adR),l ), where f , f and f are given by
2 0 1 2
f (x)=(l ,r ), f =0, f (x,y)=ω(x,y).
0 x x 1 2
Conversely, for any morphism (f ,f ,f ) from g to (h,Ξ(h),(adL,adR),l ), there is a non-abelian
0 1 2 2
extension gˆ=(g⊕h,[·,·] ) of g by h, with Z(h)=Z(ˆg)∩h, where l,r,ω are given by
(l,r,ω)
l =Pr f (x), r =Pr f (x), ω(x,y)=f (x,y).
x L 0 x R 0 2
Proof. Let ˆg=(g⊕h,[·,·] ) be a non-abelianextensionofg by h with Z(h)=Z(ˆg)∩h. Then,
(l,r,ω)
l,r,ω satisfy Eqs. (11)-(17). By (11) and (12), we have l ∈ DerL(g) and r ∈ DerR(g). By (13)
x x
and (16), we have l (α)+r (α) ∈ Z(ˆg)∩h ⊂ Z(h), which implies that f (x)(α) ∈ Z(h), for all
x x 0
α∈h. Furthermore, it is obvious that r (Z(ˆg)∩h)=0. By the condition Z(h)=Z(ˆg)∩h, we get
x
r (Z(h))=0. Thus, we deduce that f (x)∈Ξ(h). Furthermore, by (14) and (15), we get
x 0
l (f (x),f (y))−f ([x,y] ) = ([l ,l ]−l ,[l ,r ]−r )=(adL ,adR )
2 0 0 0 g x y [x,y]g x y [x,y]g ω(x,y) ω(x,y)
= (adL,adR)f (x,y).
2
10